Effects of spatial autocorrelation structure of permeability on seepage through an embankment on a soil foundation

Computers and Geotechnics 87 (2017) 62–75

Contents lists available at ScienceDirect

Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Effects of spatial autocorrelation structure of permeability on seepage through an embankment on a soil foundation Lei-Lei Liu a, Yung-Ming Cheng a,⇑, Shui-Hua Jiang b, Shao-He Zhang c, Xiao-Mi Wang d, Zhong-Hu Wu e a

Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Special Administrative Region School of Civil Engineering and Architecture, Nanchang University, 999 Xuefu Road, Nanchang 330031, China c School of Geosciences and Info-Physics, Central South University, 932 Lushan South Road, Changsha 410083, China d School of Resource and Environment Engineering, Wuhan University, 129 Luoyu Road, Wuhan 430072, China e School of Mining, Guizhou University, Qianlong Road, Guiyang 550025, China b

a r t i c l e

i n f o

Article history: Received 13 October 2016 Received in revised form 16 January 2017 Accepted 9 February 2017

Keywords: Seepage Embankment Permeability Monte Carlo simulation Random field Spatial variation Autocorrelation function

a b s t r a c t Theoretical autocorrelation functions (ACFs) are generally used to characterize the spatial variation of permeability due to the limited number of site investigation data. However, many theoretical ACFs are available in the literature, and there are difficulties in selecting a suitable ACF for general cases. This paper proposes using the random finite element method to investigate the effects of ACF on the seepage through an embankment. Five commonly used ACFs—the squared exponential (SQX), single exponential (SNX), second-order Markov (SMK), cosine exponential (CSX) and binary noise (BIN) ACFs in the literature—are compared systematically by a series of parametric studies to investigate their influences on the seepage flow problem. Both stationary and non-stationary random fields are considered in this study. The results show that the commonly used SQX and SNX ACFs may overestimate and underestimate the seepage flow rate, respectively. It is also known that the maximum exit gradient associated with the SNX ACF is larger than those obtained using the other four ACFs. Additionally, it is proved that the deterministic approach-based design is on the conservative side and tends to be too conservative when dealing with soils with greater variation in the properties. It is also found that the SQX ACF has a higher probability of providing a more conservative design in practice. Overall, the differences between different ACFs are not significant and are within acceptable ranges. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Accurate estimation of the seepage flow through a soil embankment is important towards the assessment of the safety of an embankment. In general, deterministic approaches that consider the soil permeability as a constant for a specific soil layer are employed to perform the seepage analysis [1]. However, due to the depositional and post-depositional processes, the soil permeability generally varies from point to point, even in a ‘‘homogeneous” soil layer. Such uncertainty (i.e., spatial variation) of the soil permeability should be explicitly incorporated into the seepage analysis model. To date, increasing attention has been paid to the probabilistic analysis of seepage considering the spatial variation of soil perme-

⇑ Corresponding author. E-mail addresses: [email protected] (L.-L. Liu), [email protected]. hk (Y.-M. Cheng), [email protected] (S.-H. Jiang), [email protected] (S.-H. Zhang), [email protected] (X.-M. Wang), [email protected] (Z.-H. Wu). http://dx.doi.org/10.1016/j.compgeo.2017.02.007 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.

ability [1–8], since the pioneering works by Griffiths and Fenton [9] and Fenton [10]. For example, Fenton and Griffiths [4,6] investigated the effects of the spatial variation of soil permeability on the statistics of seepage through an earth dam using the random finite element method (RFEM). Before long, Griffiths and Fenton [3,5] studied the stochastic nature of the steady seepage beneath a single sheet pile wall embedded in a spatially variable soil using three-dimensional (3D) finite element analysis and the Monte Carlo simulation (MCS). Gui et al. [1] employed a probabilistic approach to explore the effects of seepage on the slope stability of an embankment, where the hydraulic conductivity was modelled as a spatially stationary random field following a lognormal distribution. Ahmed [7] proposed combining MCS with anisotropic random fields to investigate the free surface flow through earth dams. Srivastava et al. [8] quantified the influence of the spatial variability of permeability on the steady-state seepage flow and slope stability analysis. Cho [2] developed a probabilistic seepage analysis approach that accounted for the uncertainties and spatial variation of the hydraulic conductivity in a layered soil profile, and

63

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

an earth embankment on a soil foundation was taken as an example to investigate the effects of the globally non-stationary random field of permeability on the steady seepage flow. According to all these studies, it is found that the spatial variability of permeability has been explicitly considered in steady seepage analysis during recent years. However, these previous works still suffer from some serious deficiencies, which should be addressed. The deficiencies include the following: (1) most of the works modelled the spatial variation of permeability as an isotropic random field, where the scale of fluctuation (SOF) in the horizontal direction was the same as that in the vertical direction. Nevertheless, in reality, the soil properties fluctuate more in the vertical direction than the horizontal direction due to the natural stratification and deposition of soil deposits; (2) the random fields underlying the soil permeability are commonly assumed to be globally stationary because only a ‘‘homogeneous” soil layer was considered in most of these studies, except for the works by Cho [2]. However, multiple soil layers are commonly found in practice, where the stationary random field is no longer applicable. Instead, the hydraulic conductivities are globally non-stationary; and (3) the limitation in using the theoretical single exponential (SNX) autocorrelation function (ACF), which are commonly employed to characterize the spatial variation of permeability when site investigation data are limited. However, many other theoretical ACFs are available in the literature and may provide different simulation results that are critical to the safety property and geotechnical design of the embankment. Hence, the results obtained from different ACFs should be compared systematically. The major objective of this study is to investigate the steady seepage through an embankment on a soil foundation using RFEM. The random permeability field was simulated using the extended Cholesky decomposition technique. Both situations of stationary and non-stationary random fields considering anisotropic heterogeneity are investigated. Five commonly used ACFs—squared exponential (SQX), SNX, second-order Markov (SMK), cosine exponential (CSX) and binary noise (BIN) ACFs—are summarized in Table 1 [11]. These ACFs were investigated systematically to explore the effects of the various ACFs on the steady seepage through an embankment. Hence, the present work will address the three limitations of the present development as mentioned in the previous paragraph. To achieve these objectives, the rest of this paper is organized as follows. Section 2 introduces the deterministic and stochastic seepage analyses in the two-dimensional (2D) domain, followed by the discretization of the stationary and non-stationary random fields to characterize the spatial variation of permeability using the extended Cholesky decomposition technique in Section 3. In Section 4, an embankment on soil foundation is taken as an illustrative example to investigate the effects of various ACFs on the seepage through the whole system based on stationary and nonstationary random fields. The conclusions are drawn in Section 5.

Table 1 Common ACFs for characterizing the spatial variation of permeability. ACF types SQX SNX SMK CSX BIN

ACF expressions in 2D domain 
 2 s2 qðsx ; sy Þ ¼ exp p sd2x þ d2y v h h  i qðsx ; sy Þ ¼ exp 2 dshx þ dsvy h  i   qðsx ; sy Þ ¼ exp 4 dshx þ dsvy 1 þ 4dshx 1 þ 4dsvy h  i     qðsx ; sy Þ ¼ exp  sdhx þ dsvy cos sdhx cos dsvy (   s 1  sdhx 1  dvy for sx 6 dh and sy 6 dv qðsx ; sy Þ ¼ 0 otherwise

2. Seepage analysis 2.1. Deterministic analysis based on FEM The steady seepage problem in the 2D domain is governed by a Laplace’s equation that is derived based on the assumption that the saturated-unsaturated flow obeys Darcy’s law [12]. In Cartesian coordinates, the equation is written as:



 @ @h @ @h þ ¼0 kx ky @x @x @y @y

ð1Þ

where h is the piezometric head that is equal to the summation of the pressure head and the elevation head, and kx and ky are the hydraulic conductivities in the horizontal and vertical directions, respectively. For saturated–unsaturated flow, the hydraulic conductivity depends highly on the degree of saturation or the matric suction in unsaturated soils. In general, the hydraulic conductivity function can be estimated by empirical and semi-empirical expressions in the literature. Following Van Genuchten [13], however, a set of closed-form equations as given in Eqs. (2) and (3) were used in this study to describe the soil hydraulic conductivity functions as

Se ¼

h  hr 1 ¼ ; hs  hr ½1 þ ðawÞn m


 n1 ;n > 1 m¼ n

m 2

Þ  k ¼ ks Se1=2 ½1  ð1  S1=m e

ð2Þ

ð3Þ

where Se is the effective water saturation; hr is the residual volumetric water content, hs is the saturated volumetric water content; w is the matric suction or the negative pore water pressure; m, n and a are retention parameters; k is the hydraulic conductivity function; and ks is the saturated hydraulic conductivity. To solve Eq. (1), numerical methods, such as FEM and FDM (finite element and difference methods), are commonly adopted in the literature. In the present study, an iterative FEM was utilized to obtain the numerical results, which terminates when the difference between the results from successive iterative steps is within the predefined limit. The specific solution process of this method can be found in Fredlund and Rahardjo [14] and Cho and Lee [15]. Once the equation is solved, the seepage outputs relating to the flow rates and exit gradients can be easily obtained. 2.2. Probabilistic analysis based on MCS It is preferable to perform probabilistic seepage analysis due to the stochastic nature of permeability. In the current study, the spatial variation of permeability was considered and incorporated into the FEM model to establish the stochastic analysis model. The spatial variation of the permeability was simulated based on a 2D random field generator, which will be illustrated later. MCS was then performed to repeat the stochastic analysis model analysis, based on which the statistics of seepage outputs, such as the means and standard deviations, are easily obtained. These results can provide more insights into the seepage through the underlying system. 3. Characterization of spatial variation of permeability 3.1. Simulation of random permeability field Random field theory has been widely used to characterize the spatial variation of soil properties [16–18]. Within the framework of random field theory, the soil parameters at particular locations are often considered as random variables. The resultant random field is considered stationary or weakly stationary when the

64

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

combination of the following three requirements is met [11,19–21]: (1) the statistics (i.e., means and standard deviations) of these random variables are constant over the domain of the random field; (2) the covariance between any two random variables at two points depends only on their absolute distances but not on their locations; and (3) the probability density function (PDF) for the same number of random variables has nothing to do with their absolute locations. Otherwise, the field is non-stationary. A stationary random field is usually applied to model the spatial variability of soil parameters in a nominally ‘‘homogeneous” soil layer, whereas a nonstationary random field is suitable for multi-layered soils [11]. Regarding the simulation of non-stationary random fields, the simulation domain must be divided into several non-overlapping subdomains. Soil properties at any two points in different subdomains are assumed to be uncorrelated. However, the soil parameters in each subdomain are characterized by stationary or weakly stationary random fields, where the correlation between any two points depends merely on their absolute distance instead of the locations. Hence, a non-stationary random field is easily realized based on the generation of the stationary random field in each subdomain. For details, the reader can refer to the works by Cho [22], Lu and Zhang [23] and Jiang and Huang [21], and only the simulation process of stationary random fields is described herein. To simulate a stationary random field of permeability, the Cholesky decomposition technique [11,24] is employed in this study because this method is conceptually simple and easily implemented. Generally, in a stationary random field, the correlation between any two random variables at different points is usually characterized by an ACF, which is very difficult to obtain due to the limited site investigation data. Instead, theoretical ACFs are used as alternatives [11,25]. For example, the SNX ACF is the most commonly used ACF in probabilistic seepage analysis due to its simplicity [2,4–6]. It has also been widely used to characterize the inherent spatial variation of soil properties in slope stability analysis (e.g., [18,22,26,27]). Another ACF, named SQX, is also frequently used to characterize the spatial variation of hydraulic conductivity, such as in the work by Gui et al. [1], and the resultant random field is relatively smooth. Additionally, there are some other ACFs such as CSX and BIN, which are also often used to estimate the SOFs of different soils in practice with good fitness results [11,28,29]. However, how the seepage outputs would change when different ACFs are used in the seepage problem is still an open question. Hence, five ACFs are selected herein to characterize the autocorrelation structure of permeability and to investigate the effects of these ACFs on the seepage analysis. The expressions of the five ACFs are listed in Table 1, where sx ¼ jxi  xj j and sy ¼ jyi  yj j are the absolute distances between two points in the horizontal and vertical directions, respectively, and dh and dv are the horizontal and vertical SOFs of permeability, respectively. Assume a subdomain is discretized into ne random field elements (i.e., ne random variables), and the centroid coordinates of each element is ðxi ; yi Þ, based on which the autocorrelation matrix C ne ne is expressed as

2

C ne ne

3    qðsx1ne ; sy1ne Þ 1    qðsx2ne ; sy2ne Þ 7 7 7 7 .. .. .. 7 . . . 5 qðsxne 1 ; syne 1 Þ qðsxne 2 ; syne 2 Þ    1

1 6 qðs ; s Þ 6 x21 y21 6 ¼6 . 6 .. 4

qðsx12 ; sy12 Þ

ð4Þ

where qðsxij ; syij Þ denotes the autocorrelation coefficient between spatial quantities at any two points, in which the lags sxij ¼ jxi  xj j and syij ¼ jyi  yj j denote the absolute distances between the centroid coordinates of the ith element and jth element in the horizontal and vertical directions, respectively. The

Cholesky decomposition technique is then utilized to factor the autocorrelation matrix C ne ne as

LLT ¼ C ne ne

ð5Þ

where L is an ne  ne lower triangular matrix. A typical realization of the standard Gaussian random field of permeability is then derived as:

X i;G ðx; yÞ ¼ Lni ; ði ¼ 1; 2; . . . ; NÞ

ð6Þ

where the superscript G means the standard Gaussian random field; ni is an ne  m sample matrix, which is obtained by rearranging a vector of ns independent standard normal samples as m vectors with a dimension of ne ; and i is the number of realization of the standard Gaussian random field. Thereafter, a realization of the non-Gaussian random field can be achieved by isoprobabilistic transformation as

X i;NG ðx; yÞ ¼ F 1 fU½X i;G ðx; yÞg;

ði ¼ 1; 2; . . . ; NÞ

ð7Þ

where the superscript NG means the non-Gaussian random field, F 1 ðÞ is the inverse function of the marginal cumulative distribution of permeability, and UðÞ is the standard Gaussian cumulative distribution function. Repeating the above procedure N times will give N simulations of the stationary random field, based on which the probabilistic seepage analysis can be performed. 3.2. Comparison of ACFs in graph To gain more insight into the differences among the different ACFs provided in Table 1, this subsection compared the graphs of these ACFs for the baseline case where dh ¼ 40 m and dv ¼ 4 m, as shown in Fig. 1. In this figure, as mentioned above, sx and sy are the two independent variables in the horizontal and vertical directions, respectively, while the autocorrelation coefficient qðsx ; sy Þ is the corresponding dependent variable. Generally, although qðsx ; sy Þ varies between 0 and 0.8 in all sub-graphs, there are significant differences among them. For example, the function surfaces associated with SQX and SMK ACFs are isotropic and vary smoothly and slightly with sx and sy , and they are differentiable at the origin. By contrast, the other three ACFs (i.e., SNX, CSX and BIN) are not differentiable, so there is a sharp corner at the origin and four edge angles near sx ¼ 0 and sy ¼ 0 in each of their corresponding graphs. Furthermore, compared with the other ACFs, SNX decreases sharply with increasing sx and sy [11]. 4. Analysis and results 4.1. Illustrative example This section will take a hypothetical earth embankment founded on a soil layer as an example to investigate the seepage through the system. This example was also studied by Cho [2]. The geometry of the embankment is shown in Fig. 2. The embankment has a height of 5 m and a slope of 26.6° both upstream and downstream, and the soil foundation is 5 m thick. The water level in the reservoir is 4.5 m above the soil foundation. The soil parameters including the means and coefficients of variation (COVs) for seepage analysis follow those by Cho [2] and are given in Table 2. Only the saturated hydraulic conductivity ks is taken as the random field following a lognormal distribution with a mean of 1  105 m/s and a COV of 0.3, and the horizontal and vertical SOFs of ks are 40 m and 4 m, respectively. This set of parameters is considered as the baseline case of this study, whereas others are used for parametric studies in the following sections. It should be noted that in each parametric study, only one parameter is changed, whereas the others are kept the same as those in the baseline case.

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

65

Fig. 1. Graphs of the five common ACFs for the baseline case. (a) SQX. (b) SNX. (c) SMK. (d) CSX. (e) BIN.

4.2. Deterministic analysis results Using the mean values of the soil parameters listed in Table 2, the deterministic seepage analysis based on FEM is first performed to obtain the general seepage behaviour. The FEM mesh mainly consists of 4-noded quadrilateral elements that are degenerated into 3-noded triangular elements at the sloping boundary, which results in a total of 3720 elements with 3857 nodes (Fig. 2). This mesh is relatively fine to produce accurate seepage results. The FEM results are shown in Fig. 3. The seepage velocity vectors, the

hydraulic gradient contours and the phreatic surface (zero pressure head surface) at the steady state are also shown in this figure. The main outputs of interest from the seepage analysis are the total flow rate Q that passes through the embankment and the soil foundation, and the maximum exit gradient ie. Q is calculated as 1.9964  105 m3/s, and ie is 0.867 at the toe of the downstream. These values are quite consistent with the values (i.e., 1.9947  105 m3/s and 0.863, respectively) provided by Cho [2]. These results indicate the correctness and feasibility of the current FEM model established based on SEEP/W, which is then used in

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

5m

5m

4.5 m

66

6m

10 m

2m

10 m

6m

Fig. 2. Embankment geometry and FEM mesh.

Table 2 Soil parameters for seepage analysis. Parameters

Means

COVs

dh (m)

dv (m)

ks (m/s) hs hr a (kPa1) n

1  105 0.450 0.430 1.478 2.680

0.3, 0.5, 0.7, 1.0

10, 20, 40, 60

2, 4, 10, 20

Note: Symbol ‘‘” means the item is not applicable.

0.2

0.1

0.05

0.15

0.25

0. 3

Fig. 3. Deterministic FEM results.

combination with MCS to perform the probabilistic analysis in the next section. 4.3. Probabilistic analysis results From this section on, the stochastic nature of seepage will be illustrated using RFEM. Random fields were generated using the abovementioned Cholesky decomposition to describe the spatial variation of the saturated hydraulic conductivity. The discretized random field elements are consistent with the finite elements as shown in Fig. 2, so there are a total of 3720 random variables in each realization of MCS (or ne = 3720). The random variable value at the centroid of each random field element is then assigned to the corresponding finite element. The abovementioned procedure was implemented with the help of an in-house Matlab toolbox, which has been successfully applied to some previous studies (e.g., [27]). Likewise, the random field is thus incorporated into the FEM, based on which MCS can be performed to obtain the statistics of the seepage outputs Q and ie. However, it should be noted that the number of MCS samples is critical to the accuracy of the statistics of Q and ie and must be selected carefully. A larger number would result in a more accurate estimation but also cost more computational time. In general, a suitable number is determined based on the compromise between accuracy and efficiency. According to Baecher and Christian [30] and Haldar and Mahadevan [31], a minimum number of 1536 MCS samples would yield a good estimation of mean and variance

with an acceptable error of 5% and a = 0.05—i.e., 95% confidence level in the estimated statistics. In addition, a sensitivity analysis of the normalized flow rate Q/Qdet (Qdet is the deterministic flow rate value of 1.9964  105 m3/s based on the mean inputs in Table 2) with the number of MCS samples was also performed based on the baseline case. As shown in Fig. 4, both the means and standard deviations of Q/Qdet for the five ACFs remain relatively steady when the number of MCS samples is greater than 1500. Therefore, the number of 2000 was selected in the following studies. 4.3.1. Effects of ACFs on seepage based on stationary random fields This section investigated the effects of ACFs on the seepage outputs based on the stationary random fields. Herein, the stationary random fields mean that the embankment and the soil foundation are assumed to be a geological ‘‘homogeneous” soil layer so that they share the same physical-mechanical properties. As the seepage outputs are influenced by many parameters such as COV, dh and dv of permeability, various parametric studies were conducted to investigate the variations of Q and ie with respect to these parameters for different ACFs. For each parametric study as suggested above, a total of 2000 simulations were realized and evaluated to obtain a relatively accurate estimation of the statistics of the seepage outputs. Fig. 5a shows the mean values of the normalized total flow rate Q/Qdet associated with different ACFs for various values of the COV of permeability, COV ks . In general, the means associated with all

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

Fig. 4. Variations of the mean and standard deviation of the Q/Qdet with the number of MCS samples based on the baseline case. a. Mean vs. number. b. Standard deviation vs. number.

ACFs decrease with increasing COV ks , which is consistent with the observation by Cho [2]. Such an observation seems to be counterintuitive, according to some observers such as Griffiths and Fenton

67

[5]. However, it is reasonable due to the fact that seepage flow occurs anywhere in the studied domain in a continuous flow regime. Likewise, if COV ks becomes larger, the volume of the lowpermeability region through which seepage flow passes would be greater, thus reducing the flow rate. This point also agrees well with that proposed by Fenton and Griffiths [6]. In their opinion, increasing the variance of saturated permeability would increase the chance to obtain a small hydraulic conductivity, thereby resulting in a decreased mean flow rate. It is also observed from Fig. 5a that the difference in the means for the five ACFs is negligible when COV ks is very small, but it increases sharply as COV ks becomes larger. The results using SQX ACF are generally larger than those obtained by the other four ACFs, which suggests that the total flow rate calculated using SQX ACF is much more conservative. In contrast, the most commonly applied SNX ACF may lead to unconservative seepage flow rates. The significant difference in mean total flow rates associated with SQX and SNX ACFs can be easily explained by their function graphs and expressions. From Table 1, it is evident that the autocorrelation coefficient from SQX is much larger than that from SNX, which indicates that the random permeability field based on SQX can be governed by fewer effective independent random variables from the respect of averaging process, thus presenting less variation, which reduces the chance of the flow passing through low-permeability regions in the domain. However, according to Figs. 1a and b, the SQX graph is very smooth, whereas the SNX graph decreases sharply with the absolute distance between two points. This suggests that the random permeability field based on SNX would fluctuate significantly and not easily form continuous flow paths in the domain, as can also be seen in Figs. 13a and b for an intuitive feeling. Furthermore, the results by CSX and BIN ACFs are almost the same in the study, as expected from the little differences between their graphs shown in Figs. 1d and e.

Fig. 5. Effects of COV ks on the seepage outputs for different ACFs based on stationary random fields. (a) Mean Q/Qdet vs. COV ks . (b) Standard deviation of Q/Qdet vs. COV ks . (c) Mean max ie vs. COV ks . (d) Standard deviation of the max ie vs. COV ks .

68

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

Fig. 5b presents the estimated standard deviations of Q/Qdet associated with different ACFs for various values of COV ks . As seen in the figure, the standard deviations associated with the five ACFs increase with increasing COV ks , and the results associated with SNX ACF are generally smaller than those based on the other ACFs. This is understandable from the statistics that the variance of an average decreases linearly with increasing number of independent samples used in the average and increases linearly with increasing variance of the samples—i.e., VarðXÞ / 1n VarðXÞ —if one considers the seepage flow through the system as an averaging process [5]. Based on this theory, the variance of the estimated total flow rate would increase when COV ks increases for a particular ACF because the number of effective independent samples can be considered invariant since it depends mainly on the SOFs. However, it can also be concluded from the above theory that, within the framework of the random field, a larger number of effective independent samples would result in a lower variable estimate of total flow rate, and vice versa. However, as mentioned above, when the random permeability field is constructed based on SNX ACF, there would be more effective independent samples compared with other ACFs. As such, the standard deviations associated with SNX ACFs are smaller than those from others. Overall, similar to the means of Q/Qdet, the difference among the results underlying the five ACFs increases with the increase of COV ks , but it is still less than 20%. Fig. 5c compares the mean max ie associated with five ACFs for various COVs of ks. In contrast to the observations in Fig. 5a, the results underlying all ACFs increase with increasing value of COV ks . The explanation lies in the fact that the groundwater must bear some friction to form a continuous flow, continuously consuming mechanical energy and thus leading to a head loss, and the greater the head loss, the larger the gradient. Hence, with increasing variance of permeability, the chance that the water flow passes through low-permeability regions to form a continuous flow will increase accordingly, which consequently results in a greater loss of head and thus a larger mean gradient. Additionally, it is found that the difference in means of the max ie associated with all ACFs is proportional to the value of COV ks , and the difference can be neglected at small COVs. The results using SQX, CSX and BIN ACFs are very close and are smaller than those using SNX and SMK ACFs. It is also noted that the most commonly used SNX ACF may produce the most conservative results. The rationale is that upon using SNX ACF, the resultant random permeability field may be characterized by more effective independent random variables from the point view of the averaging process, thus leading to larger variation in the permeability. This finally increases the chance of flow passing through low-permeability regions while consuming more mechanical energy, so the estimated mean value of max ie is expected to increase. Fig. 5d shows the standard deviation of the max ie associated with five ACFs for various COVs of ks. Similar to Fig. 5c, the results using all ACFs increase as COV ks becomes larger. This is expected from the aforementioned statistical theory that the variance of the average increases linearly with increasing variance of the samples. In addition, there are generally small differences among different ACFs, and the difference herein is much smaller than those in Figs. 5a, b and c, even for large values of COV ks . Meanwhile, the results underlying SNX ACF may be overestimated, whereas the results using the other four ACFs are nearly the same and may be underestimated, as can be expected from the differences in their function natures shown in Fig. 1. Fig. 6a shows the mean values of the normalized total flow rate Q/Qdet associated with different ACFs for various dh . Generally, the estimated mean values of Q/Qdet for all ACFs show an increasing trend as dh increases. This is because the simulated random field is much smoother when dh is larger, so it is easier to form a continuous

flow in the seepage domain, thereby leading to an increasing flow rate at each realization. It is also showed that, similar to Fig. 5a, the seepage flow rates underlying the most commonly used SQX and SNX ACFs are overestimated and underestimated, respectively. The explanation also lies in the fact that the random permeability field based on SQX ACF is much smoother and presents less variation than that based on SNX ACF, so it is easier to flow continuously in the domain. Additionally, using the CSX and BIN ACFs may produce almost the same results, as also expected from their function shapes. Fig. 6b compares the standard deviations of Q/Qdet estimated based on different ACFs for various values of dh . Similar to Fig. 5b, the results underlying all ACFs increase with the increase of dh , and SNX ACF provides the smallest variance among all ACFs. The rationale is also very similar to that for Fig. 5b. As mentioned above, with increasing dh , the random field becomes more correlated and is represented by fewer effective independent samples, so increasing variance of Q/Qdet is to be expected. While the autocorrelation coefficient obtained from SNX is generally smaller than those based on the others, the number of resultant effective independent samples is the largest, thereby producing the smallest variance. Fig. 6c shows that mean max ie associated with five ACFs for various dh . The mean max ie estimated using SNX ACF is larger than the values calculated based on the other four ACFs, which indicates that the results obtained by SNX ACF are overestimated. This is also because the random permeability field based on SNX fluctuates much more heavily, so the consumption of the mechanical energy or the head loss is larger, thus producing a bigger gradient. Furthermore, the results underlying SQX ACF might be underestimated at large values of dh , and the results associated with CSX and BIN ACFs are very close, which is to be expected form Fig. 1. Fig. 6d shows the standard deviations of the max ie that decrease with the increase of dh , which is quite similar to the findings of Cho [2] and Griffiths and Fenton [3]. The results associated with SQX, SMK, CSX and BIN ACFs are almost the same, which are smaller than the results underlying SNX ACF. Similarly, this observation is to be expected and attributed to the function natures of these ACFs. Fig. 7a shows the mean Q/Qdet associated with different ACFs for various dv . Similar to Fig. 6a, applying SQX and SNX ACFs may produce overestimated and underestimated results, respectively. The underlying rationale is also very similar to that for Fig. 6a. The explanation is that the random field based on SQX ACF is much smoother than that based on SNX ACF, so it is easier to form a continuous flow in the simulated random field based on SQX ACF. However, the results associated with CSX and BIN ACFs are almost the same due mainly to the similar function shapes as shown in Fig. 1d and e. Furthermore, compared with Fig. 6a, it is also found that the mean Q/Qdet tends to be more sensitive with the variation of dh . This is because the principal flow in this study is in the horizontal direction, which was also noted by Cho [2]. Fig. 7b shows the standard deviations of Q/Qdet estimated based on different ACFs, which increase with increasing dv . The standard deviations associated with SQX, SMK, CSX and BIN ACFs are very close and are slightly larger than the results obtained using SNX ACF. These observations are indeed very similar to those in Fig. 6b, so they are attributed to reasons similar to those mentioned before. Fig. 7c shows that the mean max ie decreases with increasing dv for all ACFs. This is because the simulated random field based on a specific ACF would be much smoother when dv becomes larger, which consequently leads to less consumption of the mechanical energy or the head loss, thus reducing the gradient at each realization. In addition, the difference between the results associated with different ACFs is similar to that shown in Fig. 6c and is minimal. Fig. 7d shows the variations of standard deviations of max ie

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

69

Fig. 6. Effects of dh on the seepage outputs for different ACFs based on stationary random fields. (a) Mean Q/Qdet vs.dh . (b) Standard deviation of Q/Qdet vs.dh . (c) Mean max ie vs.dh . (d) Standard deviation of the max ie vs.dh .

Fig. 7. Effects of dv on the seepage outputs for different ACFs based on stationary random fields. (a) Mean Q/Qdet vs. dv . (b) Standard deviation of Q/Qdet vs.dv . (c) Mean max ie vs.dv . (d) Standard deviation of the max ie vs.dv .

associated with all ACFs, where very close results are identified between SMK, CSX and BIN, while the results obtained based on SQX and SNX ACFs are underestimated and overestimated, respec-

tively. Obviously, these observations are very similar to those shown in Fig. 6d, and thus are attributed to reasons similar to those mentioned above.

70

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

4.3.2. Effects of ACFs on seepage based on non-stationary random fields This section investigated the effects of the five ACFs on the seepage through the same embankment studied above using the non-stationary random field theory. To achieve this aim, the saturated hydraulic conductivity at any point in the upper soil layer was assumed to be uncorrelated with the saturated hydraulic conductivity at any point in the lower soil layer. On the other hand, for simplicity, the statistics of the permeability of the two layers were assumed to be the same. This suggests that the soil parameters are applicable to both the upper and lower soil layers herein. The results are shown in Figs. 8 and 9 and are described as follows. Figs. 8a and b show the means and standard deviations of the normalized flow rate Q/Qdet associated with the five ACFs for various horizontal SOFs based on non-stationary random fields, respectively. Similar to the results obtained using stationary random fields, the flow rates (in Fig. 8a) underlying SQX and SNX ACFs are overestimated and underestimated, respectively; as for Fig. 8b, the results obtained using SNX ACFs are smaller than those obtained using the other four ACFs. In addition, in both figures, the results associated with SMK, CSX and BIN ACFs show little difference, which is very consistent with the observations shown in Fig. 6a and b. These findings are to be expected due to the similar reasons for the stationary random field cases as shown in Fig. 6a and b. Fig. 8c and d show the statistics of the max ie related to the five ACFs with respect to the variation of the horizontal SOF. Again, similar to the stationary random field simulations (see Figs. 6c and d), the estimations associated with SQX, SMK, CSX and BIN ACFs are almost the same and are slightly less than the results obtained using SNX ACF. Fig. 9 shows the statistics of the seepage outputs (i.e., Q/Qdet and the max ie) using the five ACFs based on the non-stationary random field simulations for various vertical SOFs, which produces very similar observations to those shown in Fig. 7. This indicates that neglecting the correlation between different soil layers has

little influence on the difference of the seepage outputs among all ACFs. 4.3.3. Effects of ACFs on reliability-based design interpretation Deterministic approaches (e.g., FEM) using fixed permeabilities have been widely accepted by engineers in geotechnical practice to perform dam design. However, how safe is this kind of design in the end? This is a problem that should be addressed in practice. The reliability-based design (RBD), on the other hand, could be a suitable alternative. This section investigated the effects of ACFs on seepage through the embankment from the viewpoint of RBD. Regarding the RBD, it generally addresses the following two questions [3,5]: (1) what is the probability of the actual system output being larger than the deterministic result? and (2) what is the failure probability that the actual system output exceeds the critical value? However, the second question will not be addressed in this study, since it depends on many factors such as the importance of the dam. Hence, the main objective for the design of the embankment herein is to determine the probability that the true flow rate Q exceeds the deterministic result Qdet—i.e., P(Q > Qdet)—which is equivalent to determining the probability that the normalized flow rate Qdet/Q is less than unity, i.e., PðQ det =Q < 1Þ. Following Griffiths and Fenton [5], this kind of design is termed as an ‘‘unsafe” design, and the probability that Qdet underestimates the true flow rate Q should be as low as possible. Generally, it is essential to know the distribution type of Qdet/Q to estimate PðQ det =Q < 1Þ. In this study, Qdet/Q was assumed to be lognormally distributed, which is the same as that adopted by Griffiths and Fenton [5]. This assumption is also validated by Fig. 10 where the simulated results underlying all ACFs fit the lognormal distributions well. It should be noted that the mean and standard deviation for each underlying lognormal distribution are also given in this figure. These statistics were estimated directly based on the 2000 simulated seepage analyses using the formulas suggested by Griffiths and Fenton [5], to which interested readers can refer.

Fig. 8. Effects of dh on the seepage outputs for different ACFs based on non-stationary random fields. (a) Mean Q/Qdet vs.dh . (b) Standard deviation of Q/Qdet vs.dh . (c) Mean max ie vs.dh . (d) Standard deviation of the max ie vs.dh .

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

71

Fig. 9. Effects of dv on the seepage outputs for different ACFs based on non-stationary random fields. (a) Mean Q/Qdet vs. dv . (b) Standard deviation of Q/Qdet vs. dv . (c) Mean max ie vs. dv . (d) Standard deviation of the max ie vs. dv .

Fig. 10. Histograms of normalized flow rates associated with the five ACFs based on 2000 stationary random field realizations for various values of COV ks . (a) COV ks = 0.3. (b) COV ks = 0.5. (c) COV ks = 0.7. (d) COV ks = 1.0.

Fig. 11 shows the values of PðQ det =Q < 1Þ associated with the five ACFs against the variation of COV ks . Overall, the results underlying all ACFs decrease sharply with increasing COV ks . For example,

for the considered range of COV ks herein, the value of PðQ det =Q < 1Þ using SQX ACF decreases from 44.29% at COV ks ¼ 0:3 to a value of approximately 34.10% at COV ks ¼ 1:0. This suggests that the

72

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

Fig. 11. Effects of COV ks on PðQ det =Q < 1Þ.

conventional deterministic design is much safer for the more variable soils. It might be argued that the decrease of PðQ det =Q < 1Þ with the increase of COV ks is counter-intuitive. However, it is reasonable, as can be expected from Fig. 5a, which is explained as follows [5]: generally, in a continuous flow regime, as COV ks increases, the chance that seepage flow passes through low-permeability regions would be greater, thus reducing the flow rate. It is also observed that PðQ det =Q < 1Þ is less than 50% for all ACFs within the considered variation range of COV, which indicates that the deterministic design is relatively conservative. Griffiths and Fenton [5] also noted a similar observation in their works and stated that this observation is reassuring from the view point of design. Such an observation can be illustrated by the limit case where COV ks ! 0, since in this case, the true flow rate tends to be extremely close to the deterministic flow rate, which implies that the true flow rate will fall on either side of an essentially normal distribution with a very small variance and a mean of Qdet [3,5]—i.e., PðQ det =Q < 1Þ ! 50%. Actually, in geotechnical practice, permeability has been recognized as one of the most variable soil properties with a COV ranging as high as three (e.g., [32]). This further indicates that the estimation of flow rate based on the conventional deterministic approach is relatively conservative. In addition, the results obtained using SQX and SNX ACFs are overestimated and underestimated, respectively, while the results associated with the other three ACFs are moderate with little differences between them. These differences are to be expected and attributed to the same reasons as mentioned before for Fig. 5a. This indicates that it is better to use SQX ACF than to use the other ACFs to obtain a more conservative design in practice. Fig. 12a shows the influences of the horizontal SOF on PðQ det =Q < 1Þ associated with different ACFs. It is seen in this figure that the results underlying all ACFs increase with the increase of the horizontal SOF. This is because increasing the horizontal SOF would result in a much smoother random permeability field, which is much easier to form a continuous flow in a single realization. Even so, the highest probability that the deterministic flow rate Qdet is less than the ‘‘true” flow rate Q, obtained at the largest horizontal SOF, is still less than 50%. The explanation lies in the fact that when dh ! 1, the simulated random permeability field in each MCS realization is completely correlated, which can be represented by a single random permeability variable. Likewise, the resultant flow rate distribution would be identical to the permeability distribution with a mean of Qdet that is obtained based on mean permeability [3,5]. Therefore, in this case PðQ det =Q < 1Þ ! 50%. This also suggests that the deterministic approach-based design is on the conservative side because permeability is generally highly spatially variable in geotechnical practice. Similar to the observations in Fig. 11, the results obtained using the SQX and SNX ACFs are the largest and smallest, respectively. In addition, the results related to the

Fig. 12. Effects of SOF on PðQ det =Q < 1Þ. (a) Horizontal SOF dh . (b) Vertical SOF dv .

other ACFs have small differences. Similarly, these differences are to be expected and attributed to the same reasons as mentioned before for Fig. 6a. Fig. 12b shows the influences of the vertical SOF on PðQ det =Q < 1Þ associated with different ACFs. Unlike the horizontal SOF, the results underlying the five ACFs are not that sensitive to the vertical SOF, and as a whole, the results decrease slightly with the increase of the vertical SOF. This is mainly because the domain flow is in the horizontal direction. However, there are still some similarities between the influences of the horizontal and vertical SOFs. First, the results are all less than 50% for the considered range of SOF, which further validates the conservatism of the conventional deterministic approach-based design. Second, using SQX ACF may produce the most conservative design in practice, whereas using SNX ACF could be the most speculative. Finally, the differences between the results from SMK, CSX and BIN ACFs are negligible. Again, these differences are to be expected and attributed to the same reasons as mentioned before for Fig. 7a. 4.3.4. Comparison of typical realizations of random field based on different ACFs Fig. 13 compares the random fields of permeability associated with the five ACFs based on a typical (randomly selected) MCS realization for the baseline case. Note that in this realization, the random fields associated with the five ACFs were obtained based on a given set of independent standard normal variables with a dimension of 3720  1. The corresponding seepage analysis result such as the seepage flow rate passing through the vertical section is also drawn in the figure. In addition, in the figure, the random field is visualized and represented by the change of colour depth. For a specific element, the lighter the colour, the larger the permeability. Thus, the regions in light colour denote highly permeable areas. It should be noted that only the stationary random field realizations are displayed herein, and the non-stationary random fields are

73

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

a

10 9 7

6e 1.148153

6

2.02e-005 m³/sec

Elevation (m)

8

-005

1e-005

5

8.3176377e-006

4

7.943282

3

3e-006 1e-005

2 1

1.1481536e-005

0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

Horizontal distance (m)

b

10

Elevation (m)

8

2.0456e-005 m³/sec

9 -005 1.1220185e

7

7.9432823e-006

6 5

1 .2 58 92 54 e-0 0

4 3

8.9125094e-006

5

8.9125 094e-0 06

1.2589254e-005

1.25

7.9432823 e-006

2

6.30

6.3095 734e-0 06

06 7.9432823e-0

1

9573 4e-0 06

0 4e-0 8925

5 05 1.2589254e-0

7.9432823e-006

0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

22

24

26

28

30

32

34

Horizontal distance (m)

c

10

Elevation (m)

8

1.2589254e-005

7

8.9125094e-006

6 5

1.1220185e-005

4

8.9125094e-006

3 2

1.9255e-005 m³/sec

9

7.0794578e-006

1

7.0794578e-006

0 0

2

4

6

8

10

12

14

16

18

20

Horizontal distance (m) Fig. 13. Typical random field realizations with corresponding seepage analysis results based on the baseline case. (a) SQX. (b) SNX. (c) SMK. (d) CSX. (e) BIN.

neglected, since we focus only on the effects of ACFs on the characterization of the spatial variation of permeability. From Fig. 13, it is found that the generated random fields obtained using SQX and SMK ACFs are very smooth with high- or low-permeability connectivity regions, whereas the random fields obtained using the other three ACFs seem to fluctuate roughly. This is because SMK, CSX, and BIN ACFs are non-differentiable functions, whereas SQX and SMK ACFs are differentiable near the origin, as seen in Fig. 1 [11]. In addition, since the volume of low-permeability regions that pass through the considered section shown in Fig. 13c is much larger than those in the other figures, the resultant seepage flow rate is the smallest. However, the flow rate calculated using SNX ACF is the largest because there are no obvious low-permeability regions passing through the system.

5. Summary and conclusions In this paper, the RFEM has been utilized to investigate the effects of the spatial variation of permeability on seepage through an embankment on a soil foundation. The deterministic and stochastic seepage analyses were first introduced. The Cholesky decomposition technique was then adopted to discretize the random field underlying the permeability characterized by different ACFs. Since there is no guideline in choosing a suitable ACF from the published works, a systematic study is carried out in this study. Five commonly used ACFs in the literature are summarized, and they are compared systematically by a series of parametric studies to investigate their influences on seepage outputs. Both situations based on the stationary and non-stationary random fields are con-

74

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75

d

10 2.0041e-005 m³/sec

9

Elevation (m)

8 7

8.9125094e-006

6 5

1e-005

4 3

8.9125094e-006

2

7.9432823e-006

1e-005

e-006 5734 6.309

1 0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

Horizontal distance (m)

e

10

Elevation (m)

8

2.0018e-005 m³/sec

9 1e-005

7

8.912 5

094e-0 06 1e-005 1.1220185e-005

6 5 4 3

8.9125094e-006

2

7.0794578e-006

1 0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

Horizontal distance (m) Fig. 13 (continued)

sidered. Several conclusions can be drawn from this study and are described as follows: (1) Due to the function natures of different ACFs as shown in Fig. 1, the spatial variation of permeability is better to be characterized by SQX and SMK ACFs because these ACFs are differentiable near the origin and the resulting random fields are smooth, which is more consistent with the practical circumstances. In contrast, the simulated random fields using SNX, CSX and BIN ACFs are proved to be more discontinuously distributed and thus should be carefully used in practice. (2) Generally, the seepage outputs (i.e., flow rate and the max exit gradient) associated with the five ACFs are different, which depends on COV, dh and dv . However, the difference among them is small and within an acceptable range. Regarding the seepage flow rate, the commonly used SQX and SNX ACFs provide overestimated and underestimated estimations, respectively. In contrast, the maximum exit gradient associated with SNX ACF is larger than those obtained using the other four ACFs. For both the flow rate and the maximum exit gradient, the results underlying SMK, CSX and BIN ACFs are very close. These findings are applicable to the situations based on both the stationary and non-stationary random fields. (3) In the limit case where COV ks ! 0 or dh ! 1, the true flow rate tends to be extremely close to the deterministic flow rate, which implies that the true flow rate will fall on either side of an essentially normal distribution with a very small variance and a mean of Qdet [3,5]—i.e., PðQ det =Q < 1Þ ! 50%. However, in geotechnical practice, the permeability is

spatially variable and has been recognized as one of the most variable soil properties with a COV ranging as high as three (e.g. [32]). Hence, conventional deterministic design is relatively conservative. (4) In a continuous flow regime, as COV ks increases, the chance that seepage flow passes through low permeability regions would be greater, thus reducing the flow rate. Hence, it is concluded the conventional deterministic design is much safer for more variable soils. In addition, the conventional deterministic design is much safer for highly spatially variable soils, because increasing the SOF would result in a much smoother random permeability field, which is much easier to form a continuous flow in a single realization. (5) It is found that using SQX ACF has a higher chance to obtain a more conservative design in practice. In addition, compared with the effects of the vertical SOF, the RBD is more sensitive to the effects of the horizontal SOF in the current study.

Acknowledgements The work presented in this paper was supported by The Hong Kong Polytechnic University through the projects account RU3Y and G-YBHS. These financial supports are gratefully acknowledged. References [1] Gui S, Zhang R, Turner JP, Xue X. Probabilistic slope stability analysis with stochastic soil hydraulic conductivity. J Geotech Geoenviron Eng 2000;126 (1):1–9.

L.-L. Liu et al. / Computers and Geotechnics 87 (2017) 62–75 [2] Cho SE. Probabilistic analysis of seepage that considers the spatial variability of permeability for an embankment on soil foundation. Eng Geol 2012;133– 134:30–9. [3] Griffiths DV, Fenton GA. Probabilistic analysis of exit gradients due to steady seepage. J Geotech Geoenviron Eng (ASCE) 1998;124(9):789–97. [4] Fenton GA, Griffiths DV. Extreme hydraulic gradient statistics in stochastic earth dam. J Geotech Geoenviron Eng (ASCE) 1997;123(11):995–1000. [5] Griffiths DV, Fenton GA. Three-dimensional seepage through spatially random soil. J Geotech Geoenviron Eng (ASCE) 1997;123(2):153–60. [6] Fenton GA, Griffiths DV. Statistics of free surface flow through stochastic earth dam. J Geotech Geoenviron Eng (ASCE) 1996;122(6):427–36. [7] Ahmed AA. Stochastic analysis of free surface flow through earth dams. Comput Geotech 2009;36(7):1186–90. [8] Srivastava A, Sivakumar Babu GL, Haldar S. Influence of spatial variability of permeability property on steady state seepage flow and slope stability analysis. Eng Geol 2010;110(3–4):93–101. [9] Griffiths DV, Fenton GA. Seepage beneath water retaining structures founded on spatially random soil. Geotechnique 1993;43(4):577–87. [10] Fenton GA. Statistics of block conductivity through a simple bounded stochastic medium. Water Resour Res 1993;29(6):1825–30. [11] Li DQ, Jiang SH, Cao ZJ, Zhou W, Zhou CB, Zhang LM. A multiple responsesurface method for slope reliability analysis considering spatial variability of soil properties. Eng Geol 2015;187:60–72. [12] Richards LA. Capillary conduction of liquids through porous mediums. Physics 1931;1(5):318–33. [13] Van Genuchten MT. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 1980;44(5):892–8. [14] Fredlund DG, Rahardjo H. Soil mechanics for unsaturated soils. New York: John Wiley & Sons; 1995. [15] Cho SE, Lee SR. Instability of unsaturated soil slopes due to infiltration. Comput Geotech 2001;28:185–208. [16] Cho SE. Effects of spatial variability of soil properties on slope stability. Eng Geol 2007;92(3–4):97–109. [17] Jiang SH, Li DQ, Cao ZJ, Zhou CB, Phoon KK. Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation. J Geotech Geoenviron Eng (ASCE) 2014;141(2).

75

[18] Griffiths DV, Fenton GA. Probabilistic slope stability analysis by finite elements. J Geotech Geoenviron Eng (ASCE) 2004;130(5):507–18. [19] Vanmarcke EH. Probabilistic modeling of soil profiles. J Geotech Eng Div 1977;103(11):1227–46. [20] Phoon KK, Kulhawy FH. Characterization of geotechnical variability. Can Geotech J 1999;36(4):612–24. [21] Jiang SH, Huang JS. Efficient slope reliability analysis at low-probability levels in spatially variable soils. Comput Geotech 2016;75:18–27. [22] Cho SE. Probabilistic assessment of slope stability that considers the spatial variability of soil properties. J Geotech Geoenviron Eng (ASCE) 2010;136 (7):975–84. [23] Lu ZM, Zhang DX. Stochastic simulations for flow in nonstationary randomly heterogeneous porous media using a KL-based moment-equation approach. Multiscale Model Simul 2007;6(1):228–45. [24] Suchomel R, Mašín D. Comparison of different probabilistic methods for predicting stability of a slope in spatially variable c–u soil. Comput Geotech 2010;37(1–2):132–40. [25] Li KS, Lumb P. Probabilistic design of slopes. Can Geotech J 1987;24(4):520–35. [26] Wang Y, Cao ZJ, Au SK. Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet. Can Geotech J 2011;48:162–72. [27] Liu LL, Cheng YM, Zhang SH. Conditional random field reliability analysis of a cohesion-frictional slope. Comput Geotech 2017;82:173–86. [28] Cheng Q, Luo SX, Gao XQ. Analysis and discuss of calculation of scale offluctuation using correlation function method. Chin J Rock Soil Mech 2000;21(3):281–3. [29] Cafaro F, Cherubini C. Large sample spacing in evaluation of vertical strength variability of clayey soil. J Geotech Geoenviron 2002;128(7):558–68. [30] Baecher GB, Christian JT. Reliability and statistics in geotechnical engineering. Chi Chester: John Wiley Publications; 2003. [31] Haldar A, Mahadevan S. Probability, Reliability and Statistical Methods in Engineering Design. New York: John Wiley & Sons; 2000. [32] Lee IK, White W, Ingles OG. Geotechnical engineering. London, England: Pitman Publishing, Ltd.; 1983.